42 ANALYTICAL TREATMENT OF THE POLYTOPIES REGULARLY 



h 



[100] f/ 2 = 







[l'l H= 



e 9 



= ECO 





[111] 



ce, = ( 



210] j/2 = 



: r { = tO 



[2'1'1] = e x 



''.-: 



= too 



[110] f/2 = ^ 0=( 



[L'l'l] = ce x e % = 



1 | 2 



= <\r, 



[3 2 10] y/% 



— 



n - 



- 4. 



[2 2 10] j/2 = 



ce^ft 



"2 Id» i 2 



= e x (\, 



2'1'J 1 



= 



e \ e Z ClG 



['ll00]^2=« 1 C le = 6' M 



[l'l'l 1] = 



ce x e % <% 



2 1 In | 2 



= H<\t 



2 L'l'l] 



= 



«2*3 6 16 



[1110] J/2 =ca,6' 16 



[l'l'l'l] 



ce z e A 



[l'lll] 



= *fy 



3'2T1] 



— - 



e l*3*3 6 16 



[mi] =^q 6 = 6" 8 



[2'2'1'1] = 



ce l e, e§ C\ 



1 000 | 2 



= 4s 



;; 2 2 1 0] |/2 



— 



W - 



= 5. 



[2 2 2 1 0] J/2 = 



c h H fj \ 



[2 1 000 j 2 



= *i<h% 



;2Ti'i i] 



= 



H** C M 



[1 1 000] j/2 = c^i C 33 



[l'l'l'l 1Î = 



c wA 



•' 1 100" j 2 



= H<h* 



[2'1'1'1'1] 



= 



e S e é C S2 



[1 1 100] j/2 = ce, C33 



[ni'i'i] - 



ce 8 « 4 fl 



2 1 1 lu j 2 



= e i ^82 



L 3 2 I 0] j/2 



= 



e 1^3 e 3 r 32 



[U110]K2 = «%Ci a 



[3321 0] j/2 = 



ce 1 e, e i Gg 



[l'l 111] 



== $*via 



[3'2'1'L 1] 



= 



e l*2«4 6 32 



[11111] = e<? 4 C 33 



[2'2'1'1 1] = 



ce x e,e±Q % 



: 100 i 2 



= *1*3 <?83 



[3'2'1T1] 



= 



e l%^83 



[2 2 100] f/2 = c^^ 6 y 33 



[2'2'1'1'1] = 



ce 1 e s e±C< i 



:! 2 LIU | 2 



= *1*8 ( 32 



[8'2'2'1'1] 



= 



HHe* ** 



[2 2 1101 1/2 = c^ftj C' 3 o 



[2'2'2T1] = 



ce,e^C z 



[8T111Î 



= *1*4 ^33 



[4'3'2'1'11 



= h 



e iW4, C M 



[l'l'l 11] = ce^C.,1 



[3'3'2T1] = 



ce l e, e z e± C\ 



B. The characteristic numbers. 



73. From the preceding section concerned with the measure polytope 

 can be gathered the symbols with the characteristic numbers of the 

 polytopes deduced from the cross polytope, the symbols of coor- 

 dinates of which wind up in a unit, as these polytopes also belong 

 to the offspring proper of the measure polytope. So we have only 

 to add a couple of examples about polytopes, the symbols of coor- 

 dinates of which end in zero. 



Example [2110], method working from two sides 1 ). 



The number of vertices is 2 3 . 4! divided by 2!, i. e. 8. 24:2 

 = 96. 



The number of the edges passing through the pattern vertex is 

 six, for this vertex is united by edges to the vertices: 



1 210, 

 1120, 



2011, 

 2101, 



201 — 1, 

 210 — 1. 



01 . - .96.6 



So the number of edges is — - — =288. 



Z 



In order to find spaces containing limiting bodies we consider 



successively the equations: 



+ #! = 2 , + %\ + cv 2 = 3 , + Wj + œ.j + a? 3 4* # 4 = 4. 



The equations + œ i = 2 give 8 forms [110], i. e. 8 C O of vertex 

 import. 



1 ) In the two examples we omit the common factor 1/2. 



